Question: Let $V$ be the infinite region between the planes $z = -1$ and $z = 1$ oriented with outward normals that has a boundary surface $S$. Let $F(x, y, z)$ be a continuously differentiable vector field. Does the divergence theorem necessarily apply to the region $V$, the boundary surface $S$, and the vector field $F$ ? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Solution: Assume we have a simple solid region $V$ oriented with outward normals, and it has a piecewise-smooth, closed boundary surface $S$. If $F$ is a continuously differentiable vector field in $\mathbb{R}^3$, then the divergence theorem says: $ \oiint_S F \cdot dS = \iiint_V \text{div}(F) \, dV$ The region $V$ does not satisfy the conditions of the divergence theorem. It extends infinitely, which means that it can't have a closed boundary surface $S$. Therefore, we can't apply the divergence theorem in this case.